The basic arithmetic and equivalence relations of fractional difference and fractional sum were discussed in [1]}, [2]}, [3]}, [4]}. The monotonicity of fractional difference was explored in [5]}, [3]}, [7]}. The comparison principles were developed in [8]}, [9]}, [10]}, [11]}, [12]}, [13]}, which are dominant in the stability analysis of discrete time fractional order systems and derive fascinating consequences. The fractional difference inequalities were developed in [9]}, [10]}, [16]}, and the research was not comprehensive and some new inequalities were still expected, including the non-convex case, the Cauchy inequality, the Jensen inequality and the Hölder inequality, etc. Firstly, some similar properties like the classical case can be checked for the tempered case. Secondly, efforts can be made for the unsolved problems of the classical case. Thirdly, this work could try to produce some new remarkable results.
